[Thomas_H._Cormen]_Algorithms_unlocked(BookZZ.org)

Chapter 3: Algorithms for Sorting and Searching 31
Termination: If x is not in the array, then eventually the procedure gets
to the point where p and r are equal. When that happens, step 2A
computes q to be the same as p and r. If step 2C sets r to q 1,
then at the start of the next iteration, r will equal p 1, sothatp
will be greater than r. If step 2D sets p to q C 1, then at the start
of the next iteration, p will equal r C 1, and again p will be greater
than r. Either way, the loop test in step 2 will come up false, and
the loop will terminate. Because p>r, the subarray AŒp : : r will
be empty, and so the value x cannot be present in it. Taking the
contrapositive of the loop invariant (see page 22) gives us that if x is
not present in the subarray AŒp : : r, then it is not present anywhere
in array A. Therefore, the procedure is correct in returning NOT-
FOUND in step 3.
We can also write binary search as a recursive procedure:
Procedure RECURSIVE-BINARY-SEARCH.A;p;r;x/
Inputs and Output: Inputs A and x are the same as LINEAR-SEARCH,
as is the output. The inputs p and r delineate the subarray AŒp : : r
under consideration.
1. If p>r, then return NOT-FOUND.
2. Otherwise (p r), do the following:
A. Set q to b.p C r/=2c.
B. If AŒq D x, then return q.
C. Otherwise (AŒq ¤ x), if AŒq > x, then return
RECURSIVE-BINARY-SEARCH.A;p;q 1; x/.
D. Otherwise (AŒq < x), return
RECURSIVE-BINARY-SEARCH.A; q C 1; r; x/.
The initial call is RECURSIVE-BINARY-SEARCH.A;1;n;x/.
Now let’s see how it is that binary search takes O.lg n/ time on an
n-element array. The key observation is that the size r p C 1 of the
subarray under consideration is approximately halved in each iteration
of the loop (or in each recursive call of the recursive version, but let’s
focus on the iterative version in BINARY-SEARCH). If you try all the
cases, you’ll find that if an iteration starts with a subarray of s elements,
the next iteration will have either bs=2c or s=2 1 elements, depending
on whether s is even or odd and whether AŒq is greater than or less
than x. We have already seen that once the subarray size gets down to 1,